The necessity of the emphasized monotonicity assumption
in (REF ) has been one of the motivating questions of modern
research in Diophantine approximation, and there have been great
strides in the homogeneous setting. Duffin and Schaeffer
constructed a counterexample showing that monotonicity cannot be
removed when \((n,m)=(1,1)\) , and this gave birth to the
Duffin–Schaeffer conjecture [1]}, a problem
which invigorated the field for years until its eventual proof by
Koukoulopoulos and Maynard [2]}. Gallagher removed
monotonicity for \((1,m)\)  when \(m\ge 2\)  [3]}. As
mentioned above, Sprindžuk proved Khintchine–Groshev without
monotonicity for \((n,m)\)  when \(n\ge 3\) . Beresnevich and Velani
completed the homogeneous story by removing monotonicity when
\(nm>1\)  [4]}, thus settling affirmatively a
conjecture posed by Beresnevich, Bernik, Dodson, and Velani
[5]}. The work of Beresnevich and
Velani shows that monotonicity can safely be removed in all
homogeneous cases except \((n,m)=(1,1)\) , where the Duffin–Schaeffer
counterexample had already established that monotonicity could not be
removed.
